QuineMc Clusky Minimization Method Lecture L 5 3

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Quine-Mc. Clusky Minimization Method Lecture L 5. 3 Section 5. 3

Quine-Mc. Clusky Minimization Method Lecture L 5. 3 Section 5. 3

Quine-Mc. Cluskey Method • Tabular Representations • Prime Implicants • Essential Prime Implicants

Quine-Mc. Cluskey Method • Tabular Representations • Prime Implicants • Essential Prime Implicants

Tabular Representations YZ WX 00 01 11 00 !W & X 01 -- 01

Tabular Representations YZ WX 00 01 11 00 !W & X 01 -- 01 W & !Y & Z 1 -01 10 1 1 11 1 10 1 !W & Y & !Z 0 -10 F = X & Y # !W & Y & !Z # W & !Y & Z # !W & X X&Y -11 -

Prime Implicants F = X & !Y & Z # !X & !Z #

Prime Implicants F = X & !Y & Z # !X & !Z # !X & Y Each product term is an implicant A product term that cannot have any of its variables removed and still imply the logic function is called a prime implicant.

Prime Implicants X YZ 00 01 11 0 1 1 -- 10 1 1

Prime Implicants X YZ 00 01 11 0 1 1 -- 10 1 1 F = Y & !Z # X 1 -10

Prime Implicants Minterm 0 1 2 3 4 5 6 7 X O 0

Prime Implicants Minterm 0 1 2 3 4 5 6 7 X O 0 0 1 1 1 Y O 0 1 1 Z O 1 0 1 F 0 0 1 1 1 1 X YZ 00 01 11 0 1 1 -- 10 1 1 F = Y & !Z # X 1 -10

Finding Prime Implicants Step 1 2 4 5 6 7 0 1 1 1

Finding Prime Implicants Step 1 2 4 5 6 7 0 1 1 1 O 0 1 1 0 0 1 Step 3 Step 2 (2, 6) (4, 5) (4, 6) (5, 7) (6, 7) 1 1 1 0 0 1 - (4, 5, 6, 7) 1 - (4, 6, 5, 7) 1 - - All unchecked entries are Prime Implicants -10 1 -- Y & !Z X

Prime Implicants Minterm 0 1 2 3 4 5 6 7 X O 0

Prime Implicants Minterm 0 1 2 3 4 5 6 7 X O 0 0 1 1 1 Y O 0 1 1 Z O 1 0 1 F 0 0 1 1 1 1 X YZ 00 01 11 0 1 1 -- 10 1 1 F = Y & !Z # X 1 -10

Essential Prime Implicants YZ WX 00 00 1 01 11 10 1 1 1

Essential Prime Implicants YZ WX 00 00 1 01 11 10 1 1 1 1 1 Find the essential prime implicants using the Q-M method.

Essential Prime Implicants minterms YZ WX 00 00 1 01 11 10 1 1

Essential Prime Implicants minterms YZ WX 00 00 1 01 11 10 1 1 1 1 1 0 1 2 8 3 5 10 7 14 15 0000 0001 0010 1000 0011 0101 1010 0111 1110 1111

Finding Prime Implicants Step 1 0 1 2 8 3 5 10 7 14

Finding Prime Implicants Step 1 0 1 2 8 3 5 10 7 14 15 0000 0001 0010 1000 0011 0101 1010 0111 1110 1111 Step 2 (0, 1) (0, 2) (0, 8) (1, 3) (1, 5) (2, 3) (2, 10) (8, 10) (3, 7) (5, 7) (10, 14) (7, 15) (14, 15) 00000 -0 -000 00 -1 0 -01 001 -010 10 -0 0 -11 01 -1 1 -10 -111 111 - Step 3 (0, 1, 2, 3) 00 -- (0, 2, 1, 3) 00 -- (0, 2, 8, 10) (0, 8, 2, 10) (1, 5, 3, 7) (1, 3, 5, 7) -0 -0 0 --1 6 Prime Implicants 1 -10 -111 111 - 00 --0 -0 0 --1

Find Essential Prime Implicants Prime Covered Implicant minterms * 1 -10 -111 11100 --0

Find Essential Prime Implicants Prime Covered Implicant minterms * 1 -10 -111 11100 --0 -0 0 --1 0 10, 14 7, 15 14, 15 0, 1, 2, 3 X 0, 2, 8, 10 X 1, 3, 5, 7 1 2 Minterms 3 5 7 8 X 10 14 X X X X 15 X X

3 Prime Implicants F = !W & Z # W & X & Y

3 Prime Implicants F = !W & Z # W & X & Y # !X & !Z YZ WX 00 0 --1 !W & Z 00 1 01 11 !X & !Z -0 -0 10 01 11 10 1 1 1 1 111 W & X & Y

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